Problem: Integrate. $ \int \sec^2(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $-\tan(x)+C$ (Choice B) B $2\sec(x)+C$ (Choice C) C $\dfrac{2}{3}\sec^3(x)+C$ (Choice D) D $\tan(x)+C$
Solution: We need a function whose derivative is $\sec^2(x)$. We know that the derivative of $\tan(x)$ is $\sec^2(x)$. $\dfrac{d}{dx} \tan(x) = \sec^2(x)$ Because finding the integral is the opposite of taking the derivative, this means that: $ \int \sec^2(x)\,dx = \tan(x)\, + C$ The answer: $ \tan(x)\, + C$